Dynamic mortar finite element method for modeling of shear rupture on frictional rough surfaces

Abstract

This paper presents a mortar-based finite element formulation for modeling the dynamics of shear rupture on rough interfaces governed by slip-weakening and rate and state (RS) friction laws, focusing on the dynamics of earthquakes. The method utilizes the dual Lagrange multipliers and the primal–dual active set strategy concepts, together with a consistent discretization and linearization of the contact forces and constraints, and the friction laws to obtain a semi-smooth Newton method. The discretization of the RS friction law involves a procedure to condense out the state variables, thus eliminating the addition of another set of unknowns into the system. Several numerical examples of shear rupture on frictional rough interfaces demonstrate the efficiency of the method and examine the effects of the different time discretization schemes on the convergence, energy conservation, and the time evolution of shear traction and slip rate.

Appendices

Appendix A: The mortar integral matrices

The coupling matrices $\mathbf{D}_S$ and $\mathbf{M}_M$ arising from the mortar integrals are evaluated as

$$\begin{aligned} \mathbf{D}_S \left[ {j,j} \right]&=D_{jj} \mathbf{I}=\int _{\gamma _c^{\left( 1 \right) } } N_j^{\left( 1 \right) } d\gamma \mathbf{I},\quad j=1,\ldots ,n_{sl}\nonumber \\ \end{aligned}$$

(59)

$$\begin{aligned} \mathbf{M}_M \left[ {j,l} \right]&=M_{jl} \mathbf{I}=\int _{\gamma _c^{\left( 1 \right) } } \phi _j N_l^{\left( 2 \right) } d\gamma \mathbf{I},\nonumber \\&\quad j=1,\ldots ,n_{sl} ,l=1,\ldots ,n_{mas} \end{aligned}$$

(60)

While numerical integration of the mortar matrix $\mathbf{D}_S$ involves simply the integration of the slave side displacement shape functions over the current slave contact, numerical integration of the mortar matrix $\mathbf{M}_M$ is more complex because it involves the product of master side shape functions and slave side dual shape functions over the slave contact surface. To perform this integration, we follow the approach in [17, 27], in which the integration domain is discretized into contact segments, on which both shape functions are defined continuously.

Appendix B: Linearization details

An important aspect of Sect. 6.2 is the consistent linearization of (32). We supply linearization details in this section of the “Appendix”.

The linearization of the force equilibrium in (32a) is given by

$$\begin{aligned} \Delta {}^k{\mathbf{{r}}^{t + {\hat{\upalpha }}}}= & {} \Delta \left( {\frac{1}{{\beta \Delta {t^2}}}{} \mathbf{M}{}^k{\mathbf{d}^{t + {{\Delta }}t}} + {\mathbf{f}_{int}}\left( {{}^k{\mathbf{d}^{t + {{\upalpha }}}}} \right) + {\mathbf{f}_c}\left( {{}^k{\mathbf{d}^{t + {{\upalpha }}}},{}^k{{{{\varvec{\lambda }}}}^{t + {\hat{\upalpha }}}}} \right) } \right) \nonumber \\= & {} \left( {\frac{1}{{\beta \Delta {t^2}}}{} \mathbf{M} + {}^k\mathbf{{K}}_{int}^{t + {\hat{\upalpha }}}} \right) \Delta \mathbf{d} + \Delta {}^k\mathbf{f}_c^{t + {\hat{\upalpha }}} = {}^k{\mathbf{K}^{t + {\hat{\upalpha }}}}\Delta \mathbf{d} + \Delta {}^k\mathbf{f}_c^{t + {\hat{\upalpha }}}\nonumber \\= & {} \mathbf{f}_{ext}^{t + {{\upalpha }}} + \mathbf{R} - \frac{1}{{\beta \Delta {t^2}}}{} \mathbf{M}{}^k{\mathbf{{d}}^{t + {{\Delta }}t}} - {}^k\mathbf{f}_{int}^{t + {\hat{\upalpha }}} - {}^k\mathbf{f}_c^{t + {\hat{\upalpha }}} = - {}^k{\mathbf{r}^{t + {\hat{\upalpha }}}}\nonumber \\ \end{aligned}$$

(61)

where ${ }^k \mathbf{K}_{int}^{t+\hat{\upalpha }}$ is the tangent stiffness matrix and ${ }^k \mathbf{K}^{t+\hat{\upalpha } }$ is an effective stiffness matrix defined as ${ }^k \mathbf{K}^{t+\hat{\upalpha } }=\left( \frac{1}{\beta \Delta t^{2}}{} \mathbf{M}\right. \left. +{ }^k \mathbf{K}_{int}^{t+\hat{\upalpha } } \right)$. The linearization of the contact forces can be expressed as

$$\begin{aligned} \Delta {}^{{\varvec{k}}}{} \mathbf{f}_c^{t + {\hat{\upalpha }}}= & {} \left[ \mathbf{0},\Delta \left( - {}^k\mathbf{M}{{_M^{t + {{\upalpha }}}}^T}{}^k{{{\varvec{\lambda }}}}{}^{t + {\hat{\upalpha }} } \right) ,\Delta \left( {{}^k\mathbf{D}_{S}{^{t + {{\upalpha }}}}^T} {}^k{{{\varvec{\lambda }}}}{}^{t + {\hat{\upalpha }} } \right) \right] \nonumber \\= & {} \left[ {\mathbf{0},\Delta \left( { - {{\upalpha }}{}^k\mathbf{M}{{_M^{t + 1}}^T}} \right) {}^k{{{\varvec{\lambda }}}}^{t + {\hat{\upalpha }}},\Delta \left( {{{\upalpha }}{}^k\mathbf{D}{{_S^{t + 1}}^T}} \right) {}^k{{{\varvec{\lambda }}}}_{}^{t + {\hat{\upalpha }}}} \right] \nonumber \\&+ \left[ {\mathbf{0}, - {}^k{\mathbf{M}_M}^T,{}^k{\mathbf{D}_S}^T} \right] \Delta {}^k{{{\varvec{\lambda }}}}^{t + {\hat{\upalpha }}}\nonumber \\ {:=}&{{\upalpha }}{}^k \tilde{\mathbf{C}} \Delta {}^k{\mathbf{d}_{SM}} + \left[ {\mathbf{0}, - {}^k\mathbf{M}{{_M^{t + {{\upalpha }}}}^T},{}^k\mathbf{D}{{_S^{t + {{\upalpha }}}}^T}} \right] \left( {{}^{k + 1}{{{\varvec{\lambda }}}}^{t + {\hat{\upalpha }}} - {}^k{{{\varvec{\lambda }}}}^{t + {\hat{\upalpha }}}} \right) \nonumber \\= & {} - \left[ {\mathbf{0}, - {}^k\mathbf{M}{{_M^{t + {{\upalpha }}}}^T},{}^k\mathbf{{D}}{{_S^{t + {{\upalpha }}}}^T}} \right] {}^k{{{\varvec{\lambda }}}}^{t +{\hat{\upalpha }} } = - {}^k\mathbf{f}_c^{t + {\hat{\upalpha }}} \end{aligned}$$

(62)

where the matrix $\tilde{\mathbf{C}}\in {\mathbb {R}}^{\left( {2n_{sl} +2n_{mas} } \right) \times \left( {2n_{sl} +2n_{mas} } \right) }$ includes the directional derivatives of mortar matrices $\mathbf{M}_M$ and $\mathbf{D}_S$ multiplied by the current Lagrange multiplier values ${ }^k {{\varvec{\lambda }} }^{t+\hat{\upalpha } }$ and $\Delta { }^k \mathbf{d}_{SM} \in {\mathbb {R}}^{2n_{sl} +2n_{mas}}$ are the corresponding incremental displacements of slave (S) and master (M) nodes. The directional derivatives of mortar matrices $\mathbf{M}_M$ and $\mathbf{D}_S$ are given in [27].

The linearization of the contact condition in the normal direction (32b) involves the directional derivative of the gap function (35b), which is given by

$$\begin{aligned} \Delta {}^k\dot{\tilde{g}}_{nj}^{t + \hat{{\upalpha }}}= & {} - {}^k\mathbf{n}_{j}^{{t + {{\upalpha }}}^T}\left( {{}^k\mathbf{D}_S^{t + {{\upalpha }}}\left[ {j,j} \right] \Delta {}^k{{\dot{\mathbf{d}}}}_j^{t + {{\upalpha }}} - {\mathop {\sum }\limits _{l = 1}^{{n_{mas}}}} {}^k\mathbf{M}_M^{t + {{\upalpha }}}\left[ {j,l} \right] \Delta {}^k{{\dot{\mathbf{d}}}}_l^{t + {{\upalpha }}}} \right) \nonumber \\&- {{\upalpha }}\Delta {}^k\mathbf{n}_j^{{t + 1}^T}\left( {{}^k\mathbf{D}_S^{t + {{\upalpha }}}\left[ {j,j} \right] {}^k{{\dot{\mathbf{d}}}}_j^{t + {{\upalpha }}} - {\mathop {\sum } \limits _{l = 1}^{{n_{mas}}}} {}^k\mathbf{M}_M^{t + {{\upalpha }}}\left[ {j,l} \right] {}^k{{\dot{\mathbf{d}}}}_l^{t + {{\upalpha }}}} \right) \nonumber \\&- {}^k\mathbf{n}_j^{{t + {{\upalpha }}}^T}\left( {{{\upalpha }}\Delta {}^k\mathbf{D}_S^{t + 1}\left[ {j,j} \right] {}^k{{\dot{\mathbf{d}}}}_j^{t + {{\upalpha }}} - {\mathop {\sum } \limits _{l = 1}^{{n_{mas}}}} {{\upalpha }}\Delta {}^k\mathbf{M}_M^{t + 1}\left[ {j,l} \right] {}^k{{\dot{\mathbf{d}}}}_l^{t + {{\upalpha }}}} \right) ,\nonumber \\ \end{aligned}$$

(63)

where $\Delta {}_{}^k{{{\dot{\mathbf{d}}}}^{t + {{\upalpha }}}} = \frac{{\alpha \gamma }}{{\beta {{\Delta }}t}}\Delta {}^k{\mathbf{d}^{t + 1}}$ and the directional derivative of the unit normal vector ${ }^k \mathbf{n}_j^{t+1}$ is given in [27].

In the tangential direction, the directional derivatives of the contact condition (32c) with a variable friction coefficient becomes

$$\begin{aligned} \Delta _{}^kC_{tj,St}^{t + {\hat{\upalpha }} }= & {} - _{}^k\mu _j^{t + {\hat{\upalpha }} }\left( {_{}^k\lambda _{nj}^{t + {\hat{\upalpha }} } - {c_n}_{}^k\dot{\tilde{g}}_{nj}^{t + {\hat{\upalpha }} }} \right) {c_t}\Delta _{}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }}}\nonumber \\&- {}^k\mu _j^{t + {\hat{\upalpha }} }\left( {\Delta {}^k\lambda _{nj}^{t + {\hat{\upalpha }} } - {c_n}\Delta {}^k\dot{\tilde{g}}_{nj}^{t + {\hat{\upalpha }} }} \right) {c_t}{}^k\tilde{v}_{tj}^{t + {\hat{\upalpha }} }\nonumber \\&- \Delta {}^k\mu _j^{t + {\hat{\upalpha }} }\left( {{}^k\lambda _{nj}^{t + {\hat{\upalpha }} } - {c_n}{}^k\dot{\tilde{g} }_{nj}^{t + {\hat{\upalpha }} }} \right) {c_t}{}^k\tilde{v}_{tj}^{t + {\hat{\upalpha }}}\nonumber \\= & {} {}^k\mu _j^{t + {\hat{\upalpha }}}\left( {{}^k\lambda _{nj}^{t + {\hat{\upalpha }}} - {c_n}{}^k\dot{\tilde{g}}_{nj}^{t + \mathop {\hat{\upalpha }}}} \right) {c_t}{}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }}} \nonumber \\= & {} - {}^kC_{tj,St}^{t + {\hat{\upalpha }}}, \quad j \in {}^{k}St \end{aligned}$$

(64)

and

$$\begin{aligned} \Delta {}^kC_{tj,Sl}^{t + {\hat{\upalpha }} }= & {} \left| {{}^k\lambda _{tj}^{t + {\hat{\upalpha }} } + {c_t}{}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }} }} \right| \Delta {}^k\lambda _{tj}^{t + {\hat{\upalpha }} }\nonumber \\&+ \frac{{\left( {{}^k\lambda _{tj}^{t + {\hat{\upalpha }} } + {c_t}{}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }} }} \right) }}{{\left| {{}^k\lambda _{tj}^{t + {\hat{\upalpha }} } + {c_t}{}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }} }} \right| }}{}^k\lambda _{tj}^{t + {\hat{\upalpha }} }\left( {\Delta {}^k\lambda _{tj}^{t + {\hat{\upalpha }} } + {c_t}\Delta {}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }} }} \right) \nonumber \\&- {}^k\mu _j^{t + {\hat{\upalpha }} }\left( {{}^k\lambda _{nj}^{t + {\hat{\upalpha }} } - {c_n}{}^k\dot{\tilde{g}} _{nj}^{t + {\hat{\upalpha }} }} \right) \left( {\Delta {}^k\lambda _{tj}^{t + {\hat{\upalpha }} } + {c_t}\Delta {}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }} }} \right) \nonumber \\&- {}^k\mu _j^{t + {\hat{\upalpha }} }\left( {\Delta {}^k\lambda _{nj}^{t + {\hat{\upalpha }} } - {c_n}\Delta {}^k\dot{\tilde{g}} _{nj}^{t + {\hat{\upalpha }} }} \right) \left( {{}^k\lambda _{tj}^{t + {\hat{\upalpha }} } + {c_t}{}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }} }} \right) \nonumber \\&- \Delta {}^k\mu _j^{t + {\hat{\upalpha }} }\left( {{}^k\lambda _{nj}^{t + {\hat{\upalpha }} } - {c_n}{}^k\dot{\tilde{g}} _{nj}^{t + {\hat{\upalpha }} }} \right) \left( {{}^k\lambda _{tj}^{t + {\hat{\upalpha }} } + {c_t}{}^k\tilde{v} _{tj}^{t + {\hat{\upalpha }} }} \right) \nonumber \\= & {} -\left| {{ }^k \lambda _{tj}^{t+\hat{\upalpha } } +c_t { }^k \tilde{v}_{tj}^{t+\hat{\upalpha } } } \right| { }^k \lambda _{tj}^{t+\hat{\upalpha } } \nonumber \\&+{ }^k \mu _j^{t+\hat{\upalpha } } \left( {{ }^k \lambda _{nj}^{t+\hat{\upalpha } } -c_n { }^k \dot{{\tilde{g}}}_{nj}^{t+\hat{\upalpha } } } \right) \left( {{ }^k \lambda _{tj}^{t+\hat{\upalpha } } +c_t { }^k \tilde{v}_{tj}^{t+\hat{\upalpha } } } \right) \nonumber \\= & {} - {}^kC_{tj,Sl}^{t + {\hat{\upalpha }} },\quad j \in {}^kSl, \end{aligned}$$

(65)

for the sticking and slipping nodes, respectively, where the directional derivative of the weighted tangential relative velocity ${ }^k \tilde{v}_{tj}^{t+\hat{\upalpha } }$ is similar to (63) but with ${ }^k \mathbf{t}_j^{t+{\upalpha }}$ replacing $-{ }^k \mathbf{n}_j^{t+{\upalpha }}$ and $\Delta { }^k \mathbf{t}_j^{t+{\upalpha }} =\mathbf{e}_3 \times \Delta { }^k \mathbf{n}_j^{t+{\upalpha }}$. The directional derivatives of the normal and tangential components of the Lagrange multiplier are given by

$$\begin{aligned} \Delta { }^k \lambda _{nj}^{t+\hat{\upalpha } }= & {} \Delta { }^k \mathbf{n}_j^{t+{\upalpha }} \cdot { }^k {{\varvec{\lambda }} }_j^{t+\hat{\upalpha } } +{ }^k \mathbf{n}_j^{t+{\upalpha }} \cdot \Delta { }^k {{\varvec{\lambda }} }_j^{t+\hat{\upalpha } } , \end{aligned}$$

(66a)

$$\begin{aligned} \Delta { }^k \lambda _{tj}^{t+\hat{\upalpha } }= & {} \Delta { }^k \mathbf{t}_j^{t+{\upalpha }} \cdot { }^k {{\varvec{\lambda }} }_j^{t+\hat{\upalpha } } +{ }^k \mathbf{t}_j^{t+{\upalpha }} \cdot \Delta { }^k {{\varvec{\lambda }} }_j^{t+\hat{\upalpha } } , \end{aligned}$$

(66b)

where $\Delta { }^k {{\varvec{\lambda }} }_j^{t+\hat{\upalpha } } ={ }^{k+1} {{\varvec{\lambda }} }_j^{t+\hat{\upalpha } } -{ }^k {{\varvec{\lambda }}}_j^{t+\hat{\upalpha }}$.